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Model A:  Y= β_0+β_1 X+ε Model B:  Y= β_0+β_1 X+β_2 X^2+ε where Y = Earnings, X = Experience, and X Square dis Experience Squared     Suppose now that we’ve squared X and added that to the model.     Are the predictors significant in this model B? Explain.      Is this a good model? Compare this model B to model A. Which model should you use for predictive analytics purposes? Why?      Interpret the coefficients from Model B.      Can we interpret the coefficients from Model B as a causal parameters?

Model A:  Y= β_0+β_1 X+ε Model B:  Y= β_0+β_1 X+β_2 X^2+ε where Y = Earnings, X = Experience, and X Square dis Experience Squared     Suppose now that we’ve squared X and added that to the model.     Are the predictors significant in this model B? Explain.      Is this a good model? Compare this model B to model A. Which model should you use for predictive analytics purposes? Why?      Interpret the coefficients from Model B.      Can we interpret the coefficients from Model B as a causal parameters?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Model A:  Y= β_0+β_1 X+ε

Model B:  Y= β_0+β_1 X+β_2 X^2+ε where Y = Earnings, X = Experience, and X Square dis Experience Squared

    Suppose now that we’ve squared X and added that to the model.
    Are the predictors significant in this model B? Explain. 
    Is this a good model? Compare this model B to model A. Which model should you use for predictive analytics purposes? Why? 
    Interpret the coefficients from Model B. 
    Can we interpret the coefficients from Model B as a causal parameters? 

 

### Regression Basics **Data Table:** | Obs. | X | Y | Y hat | |------|---|----|-------| | 1 | 5 | 23 | | | 2 | 5 | 16 | | | 3 | 6 | 25 | | | 4 | 8 | 28 | | --- **Part I: Regression Basics** **Summary Output A** --- **Regression Statistics** - **Multiple R**: 0.97688814 - **R Square**: 0.954308 - **Adjusted R Square**: 0.939080583 - **Standard Error**: 3.277208147 - **Observations**: 5 **ANOVA** | | df | SS | |------------|----|-----------| | Regression | 1 | 672.9797 | | Residual | 3 | 32.22028 | | Total | 4 | | **Coefficients** | | Coefficients | Standard Error | t Stat | P-value | |------------|--------------|----------------|---------|----------| | Intercept | 2.531468531 | 3.604616 | 0.702285 | 0.533087 | | X | 3.43006993 | 0.433317 | 7.915839 | 0.004203 | --- **Detailed Explanation:** 1. **Data Table:** - The table presents observations with their respective values for X, Y, and the predicted \( \hat{Y} \). 2. **Regression Statistics:** - **Multiple R** is the correlation coefficient, indicating a strong positive relationship. - **R Square** quantifies the proportion of variance in the dependent variable (Y) that can be explained by the independent variable (X). - **Adjusted R Square** accounts for the number of predictors in the model. - **Standard Error** gives an estimate of the standard deviation of the error term. - **Observations** indicates the number of data points used. 3. **ANOVA (Analysis of Variance):** - Breaks down the variance into parts attributable to the model (Regression) and error (Residual). - **df**: Degrees of freedom
Transcribed Image Text:### Regression Basics **Data Table:** | Obs. | X | Y | Y hat | |------|---|----|-------| | 1 | 5 | 23 | | | 2 | 5 | 16 | | | 3 | 6 | 25 | | | 4 | 8 | 28 | | --- **Part I: Regression Basics** **Summary Output A** --- **Regression Statistics** - **Multiple R**: 0.97688814 - **R Square**: 0.954308 - **Adjusted R Square**: 0.939080583 - **Standard Error**: 3.277208147 - **Observations**: 5 **ANOVA** | | df | SS | |------------|----|-----------| | Regression | 1 | 672.9797 | | Residual | 3 | 32.22028 | | Total | 4 | | **Coefficients** | | Coefficients | Standard Error | t Stat | P-value | |------------|--------------|----------------|---------|----------| | Intercept | 2.531468531 | 3.604616 | 0.702285 | 0.533087 | | X | 3.43006993 | 0.433317 | 7.915839 | 0.004203 | --- **Detailed Explanation:** 1. **Data Table:** - The table presents observations with their respective values for X, Y, and the predicted \( \hat{Y} \). 2. **Regression Statistics:** - **Multiple R** is the correlation coefficient, indicating a strong positive relationship. - **R Square** quantifies the proportion of variance in the dependent variable (Y) that can be explained by the independent variable (X). - **Adjusted R Square** accounts for the number of predictors in the model. - **Standard Error** gives an estimate of the standard deviation of the error term. - **Observations** indicates the number of data points used. 3. **ANOVA (Analysis of Variance):** - Breaks down the variance into parts attributable to the model (Regression) and error (Residual). - **df**: Degrees of freedom
**Summary Output B** **Regression Statistics** - Multiple R: 0.978084 - R Square: 0.956648 - Adjusted R Square: 0.913296 - Standard Error: 3.909722 **Observations**: 5 --- **Coefficients Table** | Coefficients | Standard Error | t Stat | P-value | |--------------|----------------|------------|-----------| | Intercept | 8.670659 | 19.1831 | 0.451994748 | 0.695563 | | Experience | 1.886228 | 4.729599 | 0.39881345 | 0.728582 | | Experience squared | 0.080838 | 0.246166 | 0.328388911 | 0.773812 | The table above displays a regression analysis output. The **Multiple R** value of 0.978084 indicates a strong correlation between the variables. The **R Square** of 0.956648 suggests that approximately 95.7% of the variability in the dependent variable is explained by the independent variables. The **Adjusted R Square** of 0.913296 provides a more accurate representation, accounting for the number of predictors in the model. The **Standard Error** (3.909722) measures the typical distance between the actual data points and the predicted values from the model. The **Coefficients Table** lists the coefficients for each predictor, indicating how much the dependent variable is expected to increase when the predictor increases by one unit, with all other predictors held constant. The associated statistics such as **Standard Error**, **t Stat**, and **P-value** help in understanding the reliability and significance of the predictors in the model. This output could be used in an educational context to demonstrate the process and interpretation of regression analysis in statistical research.
Transcribed Image Text:**Summary Output B** **Regression Statistics** - Multiple R: 0.978084 - R Square: 0.956648 - Adjusted R Square: 0.913296 - Standard Error: 3.909722 **Observations**: 5 --- **Coefficients Table** | Coefficients | Standard Error | t Stat | P-value | |--------------|----------------|------------|-----------| | Intercept | 8.670659 | 19.1831 | 0.451994748 | 0.695563 | | Experience | 1.886228 | 4.729599 | 0.39881345 | 0.728582 | | Experience squared | 0.080838 | 0.246166 | 0.328388911 | 0.773812 | The table above displays a regression analysis output. The **Multiple R** value of 0.978084 indicates a strong correlation between the variables. The **R Square** of 0.956648 suggests that approximately 95.7% of the variability in the dependent variable is explained by the independent variables. The **Adjusted R Square** of 0.913296 provides a more accurate representation, accounting for the number of predictors in the model. The **Standard Error** (3.909722) measures the typical distance between the actual data points and the predicted values from the model. The **Coefficients Table** lists the coefficients for each predictor, indicating how much the dependent variable is expected to increase when the predictor increases by one unit, with all other predictors held constant. The associated statistics such as **Standard Error**, **t Stat**, and **P-value** help in understanding the reliability and significance of the predictors in the model. This output could be used in an educational context to demonstrate the process and interpretation of regression analysis in statistical research.
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